Time encoding and decoding of a signal

ABSTRACT

Analog signals can be fully encoded as an asynchronous time sequence generated by a time encoding machine (TEM). With knowledge of the parameters of the time encoding machine, the asynchronous time sequence can be decoded using a non-linear time decoding machine. In one embodiment, the non-linear time decoding machine generates a set of weighted Dirac delta functions centered in the intervals of the asynchronous time sequence of the time encoding machine output. The weighting of each of the delta functions is determined by the parameters of the TEM as well as the values of the time sequence. The generation of the series of delta functions is a nonlinear operation ( 500, 510, 520 ). The input signal can be recovered from the series of weighted delta functions by ( 530 ) applying this series through an impulse response filter.

CROSS-REFERENCE TO RELATED APPLICATION

The present application is a continuation of International PatentApplication No. PCT/US03/033869, filed Oct. 27, 2003, published on May6, 2004 as International Patent Publication No. WO 04/039021, whichclaims priority from U.S. Provisional Patent Application Nos. 60/421,474filed Oct. 25, 2002 and 60/421,675 filed Oct. 28, 2002, the disclosuresof which are incorporated herein by reference in their entirety, andfrom which priority is claimed.

FIELD OF THE INVENTION

The present invention relates generally to signal processing and moreparticularly relates to circuits and methods for recovering a signalfrom the output of a time encoding machine which maps amplitudeinformation into an asynchronous time sequence.

BACKGROUND OF THE INVENTION

Most signals in the world in which we live are analog, i.e., cover acontinuous range of amplitude values. However, most computer systems forprocessing these signals are binary digital systems. Generally,synchronous analog-to-digital (A/D) converters are used to captureanalog signals and present a digital approximation of the input signalto a computer processor. That is, at precise moments in timesynchronized to a system clock, the amplitude of the signal of interestis captured as a digital value. When sampling the amplitude of an analogsignal, each bit in the digital representation of the signal representsan increment of voltage, which defines the resolution of the A/Dconverter. Analog-to-digital conversion is used in numerousapplications, such as communications where a signal to be communicatedmay be converted from an analog signal, such as voice, to a digitalsignal prior to transport along a transmission line. Applyingtraditional sampling theory, a band limited signal can be representedwith a quantifiable error by sampling the analog signal at a samplingrate at or above what is commonly referred to as the Nyquist samplingrate.

While traditional A/D conversion techniques have been effective,techniques based on amplitude sampling have limitations. For example, ithas been a continuing trend in electronic circuit design to reduce theavailable operating voltage provided to integrated circuit devices. Inthis regard, over the last decade, power supply voltages have decreasedfrom five volts to three volts and there remains a desire to reduce thisfurther, such as to one volt or less. While digital signals can bereadily processed at the lower supply voltages, traditional synchronoussampling of the amplitude of a signal becomes more difficult as theavailable power supply voltage is reduced and each bit in the A/D or D/Aconverter reflects a substantially lower voltage increment. Thus, thereremains a need to develop circuits and methods for performing highresolution A/D and D/A conversion using substantially lower power supplyvoltages which will be desired in future designs.

In contrast to sampling a signal using synchronous A/D converters,circuits are known for performing asynchronous time encoding of asignal. One such circuit, referred to as an asynchronous sigma deltamodulator (ASDM) is disclosed in U.S. Pat. No. 6,087,968 to Roza (“the'968 patent”). An example of such an ASDM is illustrated herein inFIG. 1. The ASDM generates an asynchronous duty cycle modulated squarewave at output z(t) which is representative of the input signal, x(t).In the '968 patent, an ASDM is used to form an analog-to-digitalconverter by providing the output of the ASDM to a sampler and a lineardecimating filter which is sampled at a rate above the bandwidth of theinput signal. A problem with this approach is that the ASDM introducesnon-linearities in z(t) which cannot be recovered using a lineardecimation filter. As a result, the degree of signal recovery islimited. Another shortcoming with this approach is that the output ofthe ASDM must be over sampled by a high frequency clock. As thebandwidth of the input signal increases, the clock frequency alsoincreases. Even if the desired clock rate can be achieved, such highfrequency clocks demand significant power consumption.

While the '968 patent discloses the use of an ASDM to time encode ananalog signal, certain characteristics of the ASDM circuit were notpreviously appreciated. For example, the '968 patent does not disclose amethod of designing the ASDM in order to have the ASDM output signal befully invertible, i.e., allow for theoretically perfect recovery of theinput signal. Further, the '968 patent does not disclose that the ASDMis a non-linear system and that a non-linear recovery method is requiredto fully take advantage of this circuit.

OBJECTS AND SUMMARY

It is an object of the present invention to provide circuits and methodsto encode and decode a band limited signal using asynchronous processes.

It is another object of the present invention to provide non-linearoperations for recovering an input signal from the output of a timeencoding machine, such as an ASDM or integrate and fire neuron circuit.

It is an object of the present invention to provide an improved methodof recovering an input signal from the output of an asynchronous sigmadelta modulator (ASDM).

In accordance with the present invention, a method is provided forrecovering an input signal x(t) from the output signal of a TimeEncoding Machine (TEM) which provides a binary asynchronous timesequence in response to a bounded input signal x(t). The method includesreceiving the asynchronous time sequence from the TEM and measuring thetransition times, such as the time of zero crossings, of the TEM outputsignal. From the TEM transition times, a set of weighted impulsefunctions is generated. The generation of weighted impulse functions canbe computed, such as by applying an algorithm that resolves a non-linearrecursive relationship or matrix form non-linear relationship. The inputsignal can be recovered by applying the set of weighted impulsefunctions to an impulse response filter. If the transition times areexactly known, the TEM is fully invertible and the input signal can beexactly recovered by the present methods. If the transition times arequantized, the quantization level determines the accuracy of recovery ina manner that is analogous to conventional amplitude sampling.

In one embodiment, the weighted impulse functions are a set of weightedDirac delta functions which are centered in the time interval ofsuccessive zero crossings of the TEM output. The weighting value foreach impulse function is related to the design parameters of the TEM andthe transition times. For example, the weighted impulse functions canhave weighting value coefficients c_(k) which can be expressed in matrixform as a column vector, c:c=G⁺q

where q is a column vector and

∫_(tk)^(t_(k + 1))x(u)𝕕u = (−1)^(k)[−b(t_(k + 1) − t_(k)) + 2κδ]q = [∫_(t_(k))^(t_(k + 1))x(u)𝕕u]  and  G = [∫_(t₁)^(t_(l + 1))g(u − s_(k))𝕕u]  and  where  G⁺denotes the pseudo-inverse of matrix G. The input signal x(t) can berecovered from the vector c by passing a train of impulse functionsweighted by this vector through an ideal impulse response filter. Theimpulse response filter can be described as g(t)=sin(Ωt)/πt. Therecovery of the input signal can then be expressed in matrix form as:x(t)=gc, where g is a row vector, g=[g(t−s_(k))]^(T).

The TEM can take on many forms so long as the transition times from theTEM output relate to the input signal in an invertible manner. Examplesof TEM circuits include an asynchronous sigma delta modulator circuitand an integrate and fire neuron circuit.

The methods of the present invention can be embodied in a time decodingmachine (TDM) circuit. The TDM will generally be formed using a digitalmicroprocessor or digital signal processing integrated circuit which hasbeen programmed to carry out the methods of the present invention.

BRIEF DESCRIPTION OF THE DRAWING

Further objects, features and advantages of the invention will becomeapparent from the following detailed description taken in conjunctionwith the accompanying figures showing illustrative embodiments of theinvention, in which:

FIG. 1 is a block diagram of a known circuit embodiment of anasynchronous sigma delta modulator (ASDM), which can be used as anembodiment of a time encoding machine (TEM) in the present invention;

FIG. 2A is a simplified block diagram illustrating a system inaccordance with the present invention including a time encoding machine(TEM) for asynchronous time encoding of an input signal and a timedecoding machine for subsequently recovering the input signal from theTEM output signal;

FIGS. 2B and 2C pictorially represent the operation of the ASDM of FIG.1 operating as a time encoding machine where FIG. 2B is a graphicalrepresentation of an analog input signal, x(t), to the system of FIG. 2Aand FIG. 2C is a graphical representation of the time encoded signalz(t) from the time encoding machine;

FIG. 3 is a simplified flow chart illustrating the operations of thetime encoding machine (TEM) and time decoding machine (TDM) of thepresent invention;

FIG. 4 is a block diagram of a circuit including a TEM and oneembodiment of a quantizer for the TEM output signal.

FIG. 5 is a pictorial representation of the operation of the timedecoding machine;

FIG. 6 is a timing diagram illustrating the operation of a TEM for a DCinput signal; and

FIGS. 7A and 7B are timing diagrams illustrating the operation of a TEMin response to a time varying input signal, x(t);

FIG. 8 is a graph illustrating recovery error versus δ for a compensatedrecovery algorithm in accordance with the present invention.

FIG. 9 is a block diagram of an alternate embodiment of a TEM circuitusing an integrate and fire neuron circuit to time encode an inputsignal; and

FIG. 10 is a graph illustrating an example of the output signal from theTEM of FIG. 9.

Throughout the figures, the same reference numerals and characters,unless otherwise stated, are used to denote like features, elements,components or portions of the illustrated embodiments. Moreover, whilethe subject invention will now be described in detail with reference tothe figures, it is done so in connection with the illustrativeembodiments. It is intended that changes and modifications can be madeto the described embodiments without departing from the true scope andspirit of the subject invention as defined by the appended claims.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

The present invention uses time encoding to digitize an analog signaland also uses appropriate non-linear decoding methods for subsequentrecovery of the input signal. Thus, the systems and methods describedapply to analog-to-digital (A/D) conversion and digital-to-analog (D/A)conversion using time encoding of a signal.

In the present application, the concepts of a time encoding machine(TEM) and time decoding machine (TDM) are introduced. As used herein, aTEM is a circuit or process which receives a bounded input signal andgenerates an asynchronous set of binary transitions, the timing of whichare related to the amplitude of the input signal. More generally, a TEMis a circuit or process which maps amplitude information into timeinformation. A time decoding machine is a circuit or process which isresponsive to transition time data from a TEM and can process thetransition time data to recover the input signal with arbitraryaccuracy.

The block diagram of FIG. 2A provides a simplified description of thepresent invention. A band limited, amplitude limited signal x(t), asshown in FIG. 2B, is applied to the input of a Time Encoding Machine(TEM) 205 which generates a binary, asynchronous output z(t),illustrated in FIG. 2C. The TEM output signal z(t) is an asynchronousbinary output whose zero crossings (t_(o), t₁, . . . t_(k)) are relatedto the amplitude of the input signal. It is now understood in connectionwith the present invention that the signal z(t) relates to the inputsignal x(t) in a non-linear fashion. By using the inverse of thenon-linear relationship in a decoding algorithm in a Time DecodingMachine (TDM) 210, the input signal, x(t), can be recovered fromknowledge of the zero crossings in z(t). If the zero crossings in z(t)are known precisely, then perfect recovery of the input signal ispossible. If there is error attributable to the measurement of the zerocrossings, the error in the recovery of x(t) is equivalent to magnitudeof the error in the recovery of a signal using traditional amplitudesampling. That is, for the same number of bits used in sampling thesignal, the current time encoding methods yield comparable results toknown amplitude sampling systems.

FIG. 1 is a block diagram of a known Asynchronous Sigma Delta Modulator(ASDM) circuit which is suitable for use in one embodiment of TEM 205.The circuit of FIG. 1 generates a two-state output signal (b, −b) whosezero crossings are asynchronously determined by the amplitude of theinput signal x(t) in combination with design parameters of the circuit.Because the timing of the zero crossings is asynchronously determined bythe amplitude of the input signal, the ASDM can be viewed as a form oftime encoding machine (TEM). If the input signal x(t) to the TEM is aband limited signal, then the system is invertible and the input signal,x(t), can be recovered from the TEM output signal, z(t), using anappropriate recovery process in the TDM. If the transition times areknown exactly, the input signal can be fully recovered by the TDM.Accordingly, by knowing the expected maximum bandwidth of an inputsignal, the parameters of the TEM can be designed accordingly.

The underlying operating principle of a TEM can be understood byreference to FIG. 1. In order to realize complete recovery, it isexpected that the input signal x(t) is bounded in both bandwidth, Ω, andamplitude, c. The input signal, x(t), is applied to an adder 105 whichis coupled to the input of an integrator circuit 110. The integratorcircuit 110 has an integration time constant, K. The output of theintegrator circuit 110 is coupled to the input of a non-invertingSchmitt trigger 115. The Schmitt trigger has two design parameters, theoutput voltage swing (b, −b) and the hysteresis value, δ. The output ofthe Schmitt trigger 115 provides a TEM output signal, z(t). The outputof the Schmitt trigger 115 is also fed back to an inverting input to theadder 105. As a result, the bounded input signal x(t), |x(t)|≦c<b, isbiased by a constant amount ±b before being applied to the integrator110. This bias guarantees that the output of the integrator y(t) iseither a positive-increasing or negative-decreasing function of time.

In the steady state, the TEM output signal z(t) can assume one of twostates. In the first state, the TEM is in the state z(t)=−b and theinput to the Schmitt trigger increases from −δ to δ. When the output ofthe integrator circuit 110 reaches the maximum value δ, a transition ofthe output z(t) from −b to b is triggered and the feedback signalapplied to adder 105 becomes negative. In the second steady state ofoperation, the TEM is in the state z(t)=b and the integrator outputsteadily decreases from δ to −δ. When the maximum negative value −δ isreached, z(t) will reverse to −b. Thus, a transition of the output ofthe TEM from −b to b or from b to −b takes place each time theintegrator output reaches the triggering threshold δ or −δ. The timingof these transitions, which result in the zero crossings, t_(o) . . .t_(k), depends on both the amplitude of the input signal x(t) and thedesign parameters of the TEM, such as the hysteresis of the Schmitttrigger, δ. Thus, the TEM is effectively mapping amplitude informationinto timing information using a signal dependent sampling mechanism.

The Schmitt trigger 115 can be characterized by two design parameters:the output voltage swing (b, −b) and the hysteresis value, δ. In orderfor the ASDM of FIG. 1 to operate as an invertible TEM, the amplitude ofthe input signal c must be less than output voltage swing of the Schmitttrigger, i.e., |c|<b. It has also been found that to fully recover theinput signal from the TEM output signal, the hysteresis value, δ, mustbe less than the period of the Nyquist sampling rate. This can beexpressed as: δ<(π/Ω)((b−c)/κ), where Ω represents the bandwidth of theinput signal x(t), κ represents the integration constant of the TEM, andc represents the maximum amplitude of the input signal. Thus, theparameters of the TEM are to be set based on the expected limits of theinput signal.

The set of trigger times t_(k) which comprise the output of the TEM canbe expressed by the recursive equation:

$\begin{matrix}{{\int_{tk}^{t_{k + 1}}{{x(u)}{\mathbb{d}u}}} = {{\left( {- 1} \right)^{k}\left\lbrack {{- {b\left( {t_{k + 1} - t_{k}} \right)}} + {2{\kappa\delta}}} \right\rbrack}.}} & \left( {{Eq}.\mspace{14mu} 1} \right)\end{matrix}$for all integer values of k, i.e., k, kεZ. where κ is the integrationconstant of the TEM, δ is the hysteresis value of the Schmitt trigger inthe TEM and b is the amplitude output voltage swing of the TEM output.

By dividing by b on both sides of equation (2), the output can benormalized and can be expressed as:

$\begin{matrix}{{\int_{k}^{t_{k + 1}}{\frac{x(u)}{b}\ {\mathbb{d}u}}} = {{\left( {- 1} \right)^{k}\left\lbrack {{- \left( {t_{k + 1} - t_{k}} \right)} + \frac{2{\kappa\delta}}{b}} \right\rbrack}.}} & \left( {{Eq}.\mspace{14mu} 2} \right)\end{matrix}$for all k, kεZ, i.e., k is an element of the set of all integer numbers.Therefore, the increasing time sequence (t_(k)), kεZ, can be generatedby an equivalent circuit with integration constant κ=1 and a Schmidttrigger with parameters κδ/b and 1. In what follows, without any loss ofgenerality, this simplified version of the TEM will be used.

The input to the TEM is a bounded function

x(t) with |x(t)|≦c<1 for all t, tεR, i.e., t is an element of the set ofall real numbers. The output of the TEM is function z taking two valuesz: R→{−1,1} for all t, tεR, with transition times t_(k), kεZ, generatedby the recursive equation

$\begin{matrix}{{{\int_{t_{k}}^{t_{k + 1}}{{x(u)}{\mathbb{d}u}}} = {\left( {- 1} \right)^{k}\left\lbrack {{- \left( {t_{k + 1} - t_{k}} \right)} + \delta} \right\rbrack}},} & \left( {{Eq}.\mspace{14mu} 3} \right)\end{matrix}$for all k, kεZ. This equation illustrates the mapping of the amplitudeinformation of the signal x(t), tεR, into the time sequence (t_(k)),kεZ.

In the description that follows, the operation of the TEM is evaluatedfor the case where the input signal is a constant DC value. For thisexample, it is assumed that the TEM takes the form substantiallypresented in FIG. 1 and consists of an integrator with integratorconstant κ=1 and Schmitt trigger with parameters δ from −δ/2 to δ/2 andb=1. In this case, x(t)=c (not necessarily positive), where c denotes agiven DC level. If c<−1 or c>1 the output of the integrator would becomeunbounded, and thus the overall TEM would become unstable. This mightlead to information loss because the output z(t) cannot track the inputx(t). By bounding the input amplitude such that |c|<1, the TEM is stableand the sequence of transition times reduces to the simple recursion:

$\begin{matrix}{{t_{k + 1} = {t_{k} + \frac{\delta}{1 + {\left( {- 1} \right)^{k}c}}}},} & \left( {{Eq}.\mspace{14mu} 4} \right)\end{matrix}$and, therefore, t_(k+1)>t_(k), kεZ.

The integrator output, y(t), and the TEM output, z(t) for the case wherethe input signal is a constant value, are illustrated in the graphs ofFIGS. 3A and 3B, respectively. Referring to FIGS. 3A and 3B, it isapparent that both the integrator output and TEM output, y(t) and z(t),respectively, are periodic signals with a period T. The period T can beexpressed as:

$\begin{matrix}{T = {{t_{k + 2} - t_{k}} = {\frac{2\delta}{1 - c^{2}}.}}} & \left( {{Eq}.\mspace{14mu} 5} \right)\end{matrix}$for all k, kεZ.

The mean value (the 0-th order Fourier-series coefficient) of z(t) canbe expressed as:

$\begin{matrix}{{\frac{\left( {- 1} \right)^{k}\left\lbrack {{- \left( {t_{k + 1} - t_{k}} \right)} + \left( {t_{k + 2} - t_{k + 1}} \right)} \right\rbrack}{T} = c},} & \left( {{Eq}.\mspace{14mu} 6} \right)\end{matrix}$Equation 6 illustrates that the input signal and the output signal fromthe TEM have the same average value.

The transition times from the TEM output can be measured usingconventional circuits and methods. FIG. 4 is one example of a circuitfor quantizing and measuring the transition times t_(K) from the outputof a TEM. The TEM 205 has an output which is coupled to a strobe inputof a counter circuit 405. The counter 405 is also coupled to a clockcircuit 410 which increments the counter value on each clock pulse. Thusthe counter circuit 405 maintains a running count of clock pulses. Oneach occurrence of t_(K), the current counter value in counter circuit405 is strobed to an N-bit output bus of the counter circuit 405. Thecounter output can then be provided to a digital processor 410 forstorage and processing in accordance with the methods described herein.The counter values from the counter circuit 405 are transferred to theprocessor 410 at a rate equal to or greater than the Nyquist frequencyof the input signal. Thus, the timing of the zero crossings are beingquantized with a definable error based on the number of bits ofresolution in the counter 405. The processor receives the quantizedvalues of t_(k) as an N bit word from the counter 405 on each occurrenceof t_(k).

The operation of the Time Decoding Machine (TDM) 210 will be describedin connection with the flow chart of FIG. 5 and pictorial diagram ofFIG. 6. FIG. 5 is a simplified flow diagram illustrating the operationof the TDM 210. As noted above, the TEM 205 operates by generating a setof signal dependent transition times (Step 500). The transition timesare discrete real values and can be designated as t_(K)=[t₀, t₁, t₂, . .. , t_(K)]. From the set of signal dependent transition times, t_(K),the input signal x(t) can be represented as a set of weighted Diracdelta functions, S_(k). Each of the weighted delta functions has a timevalue determined by the center of each successive time interval (t_(K),t_(K+1)) and having a weighting value, c_(k), associated therewith. Forexample, referring to FIG. 6, S1 in 630 has a time value at the medianvalue between to and to which are illustrated in 620 (Step 510).

The values of the weighting coefficients can be determined from thetransition times, t_(k), and the Schmitt trigger hysteresis value, δ(step 520). The coefficients c_(k) can be expressed in matrix form as acolumn vector, c:

$\begin{matrix}{{c = {{G^{+}q{\int_{t_{k}}^{t_{k + 1}}{{x(u)}{\mathbb{d}u}}}} = {\left( {- 1} \right)^{k}\left\lbrack {{- \left( {t_{k + 1} - t_{k}} \right)} + \delta} \right\rbrack}}},{{{where}\mspace{14mu} q} = {{\left\lbrack {\int_{t_{k}}^{t_{k + 1}}{{x(u)}{\mathbb{d}u}}} \right\rbrack\mspace{14mu}{and}\mspace{14mu} G} = \left\lbrack {\int_{t_{l}}^{t_{l + 1}}{{g\left( {u - s_{k}} \right)}{\mathbb{d}u}}} \right\rbrack}}} & \left( {{Eq}.\mspace{14mu} 7} \right)\end{matrix}$where G⁺ denotes the pseudo-inverse of matrix G.

The input signal x(t) can be recovered from the vector c by passing thisvector through an ideal impulse response filter (Step 530). The impulseresponse filter can be described as g(t)=sin (Ωt)/πt. Such a filter canbe formed as a digital filter using suitable programming of processor420 (FIG. 4). It will be appreciated that other embodiments, such asanalog filters and other digital filter embodiments are alsocontemplated by the present invention. The recovery of the input signalcan be expressed in matrix form as:x(t)=gc, where g is a row vector, g=[g(t−s _(k))]^(T)  (Eq. 8)

The method steps described in connection with the flow diagram of FIG. 5are represented pictorially in FIG. 6. An input signal x(t) is appliedto the input of a TEM 610, such as a properly designed ASDM asillustrated in FIG. 1 and described above. The TEM 610 provides anoutput comprising signal dependent asynchronous timing data derived fromx(t) 620. Based on the value of the hysteresis of the TEM, δ, the timesequence of z(t) is converted into a matrix of weighted Dirac deltafunctions, S_(k) 630. The weighted Dirac delta functions have a timevalue centered between consecutive zero crossings t_(k). The weightingvalue is expressed above, c=G⁺q. The sequence S_(k) 630 can then beapplied to an ideal impulse response filter g (step 640) which recoversthe original signal, x(t).

In the method described above, perfect signal recovery can be achievedwith knowledge of the signal bandwidth, upper bound on the signalamplitude with respect to the output swing of the Schmitt trigger andthe hysteresis value of the Schmitt trigger, δ, and exact knowledge ofthe transition times, t_(k). However, in many practical cases, errorsare attributable in determining the values of the trigger times in z(t)as well as in determining the exact value of the hysteresis.

The error associated with the measurement of the trigger times resultsfrom quantization of the timing information such that the transitiontimes are not exactly known, i.e., the times are known within thequantization level used. It can be shown that the resulting error fromtime encoding is comparable to that of amplitude sampling, when the samenumber of bits of resolution are used in each case.

A reasonable comparison between the effects of amplitude and timequantization can be established if we assume that the quantizedamplitudes and quantized trigger times are transmitted at the samebitrate. Since x(s_(k)) and T_(k) are associated with the trigger timest_(k) and t_(k+1), the same transmission bitrate is achieved if thex(s_(k))'s and the T_(k)'s are represented by the same number of bits N.With −c≦x≦c, the amplitude quantization step amounts to ε=2c/2^(N).

For time encoding T_(min)=min_(κεZ)T_(k)≦T_(K)≦max_(kεZ)T_(k)=T_(max) orequivalently 0≦T_(k)−T_(min)≦T_(max)−T_(min). Therefore, if T_(min) isexactly known, then only measuring T_(k)−T_(min), kεZ, in the range(0,T_(max)−T_(min)) is needed. Hence:

$\begin{matrix}{\Delta = {\frac{T_{\max} - T_{\min}}{2^{N}} = {{\frac{1}{2^{N}}\left( {\frac{\delta}{1 - c} - \frac{\delta}{1 + c}} \right)} = \frac{\delta ɛ}{1 - c^{2}}}}} & \left( {{Eq}.\mspace{14mu} 9} \right)\end{matrix}$where Δ is the quantization step of the time sequence. Equation 9illustrates how the quantization step of the time sequence is related tothe amplitude quantization step.

The hysteresis value δ is often determined by analog components in theTEM, such as resistors and capacitors which exhibit a base linetolerance as well as variations over temperature, time and the like. Theerror that is associated with variations in the exact value of δ can besubstantially eliminated by the use of a compensation method that isdescribed below.

In the case where the TEM is embodied as an ASDM, such as illustrated inFIG. 1, following each transition of z(t), the polarity of the signalchanges. This is evident from the equations for calculating t_(k), suchas Equation 3, which includes a term (−1)^(k) which reflects this signchange on each transition, k. Since Equation 3 also includes δ for eachinstance of t_(k), it will be appreciated that by evaluating the signalover a window spanning two consecutive instances of t_(k), the δ termwill be positive in one case and negative in the next, thereby cancelingthe effect of any variation in δ over this window. The principle behindthe δ compensated algorithm can be described as follows:

${h = {\frac{1}{2}\left\lbrack {g\left( {t - t_{k + 1}} \right)} \right\rbrack}^{T}},{q = {\left\lbrack {\int_{t_{k}}^{t_{k + 2}}{{x(u)}{\mathbb{d}u}}} \right\rbrack\mspace{14mu}{and}}}$$H = {{\frac{1}{2}\left\lbrack {\int_{t_{l}}^{t_{l + 2}}{{g\left( {t - t_{K + 1}} \right)}{\mathbb{d}u}}} \right\rbrack}.}$The band limited signal x can be perfectly recovered from its associatedtrigger times (t_(k)), kεZ, as

$\begin{matrix}{{x(t)} = {\begin{matrix}{\lim\;{x_{l}(t)}} \\\left. l\rightarrow\infty \right.\end{matrix} = {{hH}^{+}q}}} & \left( {{Eq}\;.\mspace{11mu} 10} \right)\end{matrix}$where H⁺ denotes the pseudo-inverse of H. Furthermore,x _(l)(t)=hP _(l) q,where P_(l) is given by

$P_{l} = {\sum\limits_{k = 0}^{l}\;{\left( {I - H} \right)^{k}.}}$More generally, by defining a matrix B with ones on two diagonals:

$B = \begin{matrix}⋰ & \; & \; & \; & \; & \; \\{\cdots 1} & 1 & 0 & 0 & 0 & {0\cdots} \\{\cdots 0} & 1 & 1 & 0 & 0 & {0\cdots} \\{\cdots 0} & 0 & 1 & 1 & 0 & {0\cdots} \\{\cdots 0} & 0 & 0 & 1 & 1 & {0\cdots} \\\; & \; & \; & \; & \; & ⋰\end{matrix}$the input signal recovery algorithm can be expressed:x(t)=g ^(T)(t)[(B ⁻¹)(BG(B ⁻¹)^(T))⁺ [Bq]  (Eq. 11)The two diagonal row B matrix illustrated above is but one form ofprocessing which can be used. It will be appreciated that other generalexpressions of this matrix can also be applied.

The compensation δ-insensitive recovery algorithm achieves perfectrecovery. Simulation results for the δ-sensitive and δ-insensitiverecovery algorithms are shown in FIG. 8, with graph lines denoted bystars and circles, respectively.

The δ-insensitive recovery algorithm recognizes that in two consecutivetime periods of t_(k), the period of each cycle is dependent on δ andthe output changes polarity from b to −b or vice versa. By looking atthe output z(t) over two consecutive time periods, i.e., t_(n), t_(n+2),the error in the recovered version of x(t) attributable to δ caneffectively be eliminated.

FIG. 9 illustrates an alternate embodiment of a TEM. The circuit of FIG.9 can be described as an integrate and fire neuron with an absoluterefractory period, Δ. The TEM of FIG. 9 includes an adder 905 to whichan input signal x(t) is applied. The other terminal of the adder circuit905 is coupled to a fixed bias voltage, b. The adder 905 is coupled tothe input of an integrate and dump circuit 910. The integrate and dumpcircuit 910 integrates the input signal with an integration constant ofκ until a dump or reset signal is received on input 915. The output ofthe integrator is applied to a threshold detector 920 which has anoutput which is zero until the input signal exceeds the threshold level,δ. When the input voltage to the threshold detector 920 exceeds thethreshold level value the output transitions to a logic 1. The output ofthe threshold detector 920 is coupled to the dump input 915 of theintegrate and dump circuit 910. A small time delay element τ (not shown)may be included in this circuit path to provide a more defined pulse atthe threshold detector output, which will generate the transition timest_(k). The integrator constant κ, the threshold δ, the bias b in FIG. 9are strictly positive real numbers; x(t), tεR, is a signal of finiteenergy on R that models the input stimulus to the TEM. Furthermore, x isbounded, |x(t)|≦c<b, and band limited to [−Ω,Ω]. The output of theintegrator in a small neighborhood of t_(O)+Δ, t>t_(O)+Δ is given by:

$\begin{matrix}{{y(t)} = {{y\left( {t_{o} + \Delta} \right)} + {\frac{1}{k_{t_{0}}}{\int_{+ \Delta}^{t}{\left\lbrack {{x(u)} + b} \right\rbrack\ {{\mathbb{d}u}.}}}}}} & \left( {{Eq}.\mspace{14mu} 12} \right)\end{matrix}$The term Δ represents the absolute refractory period of the circuit,which is a small but finite time period between the time the integratorresets and the time the integrator begins its next integration period.The absolute refractory period alters the transfer function of theintegrator from h(t)=1 for t≧0 and h(t)=0 for all other t, to h(t)=1 fort≧Δ and h(t)=0 for all other t. Due to the bias signal b, the integratoroutput y=y(t) is a continuously increasing function. The output of theTEM is a time sequence (t_(k)), kεZ.

The output of the TEM of FIG. 9 is a strictly increasing set of triggertimes (t_(k)), kεZ, that satisfy the recursive equation

$\begin{matrix}{{{\int_{{tk} + \Delta}^{{tk} + 1}{{x(u)}\ {\mathbb{d}u}}} = {{- {b\left( {t_{K + 1} - t_{K} - \Delta} \right)}} + {\kappa\delta}}},} & \left( {{Eq}.\mspace{14mu} 13} \right)\end{matrix}$for all k, kεZ.For all input stimuli x=x(t), tεR, with |x(t)|≦c<b, the distance betweenconsecutive trigger times t_(k) and t_(k+1) is given by:

$\begin{matrix}{{\frac{\kappa\delta}{b + c} + \Delta} \leq {t_{k + 1} - t_{k}} \leq {\frac{\kappa\delta}{b + c} + \Delta}} & \left( {{Eq}.\mspace{14mu} 14} \right)\end{matrix}$for all k, kεZ.

The circuit of FIG. 9 can be generalized further by considering theintegrator circuit to be a non-ideal integrator. If the capacitor in theintegrator circuit is considered leaky, i.e., in parallel with a finiteresistance, the leaky integrator resembles a more general low passfilter response. The leaky integrate-and-fire circuit can be defined bythe transfer function:

h(t)=(1/κ)e^(−t) t≧0; h(t)=0 otherwise.

The present invention provides systems and methods for digitizing ananalog signal using time encoding rather than amplitude encoding of thesignal. The present methods also provide for recovery of the timeencoded signal with an error which is comparable to that of traditionalamplitude encoding when the sampling occurs at or above the Nyquistsampling rate. The present methods are particularly well suited for lowvoltage analog applications where amplitude sampling may be limited bythe small voltage increments represented by each bit when highresolution sampling is required with only a limited voltage supply. Theability to accurately encode analog signals using limited supplyvoltages makes the present circuits and methods well suited to use innano-scale integration applications.

Although the present invention has been described in connection withspecific exemplary embodiments, it should be understood that variouschanges, substitutions and alterations can be made to the disclosedembodiments without departing from the spirit and scope of the inventionas set forth in the appended claims.

1. A method of recovering an input signal from an output signal of aTime Encoding Machine (TEM), the TEM providing a binary asynchronoustime sequence in response to a bounded input signal, comprising:measuring times of zero crossings indicating transitions of the TEMoutput signal of the Time Encoding Machine; generating a set of weightedimpulse functions based upon the measured times of the zero crossings;and applying the set of weighted impulse functions to an impulseresponse filter to recover the input signal.
 2. A method of recoveringan input signal from the output signal of a Time Encoding Machineaccording to claim 1, wherein consecutive zero crossings of the TEMoutput signal define time intervals and wherein the weighted impulsefunctions have time values that are centered on the time intervals ofthe TEM output signal.
 3. A method of recovering an input signal fromthe output signal of a Time Encoding Machine according to claim 1,wherein the weighted impulse functions have a weighting value determinedby time values of at least two zero crossings and at least one propertyof the TEM.
 4. The method of recovering an input signal from the outputsignal of a Time Encoding Machine according to claim 3, wherein theweighted impulse functions have weighting value coefficients c_(k) whichare expressed in matrix form as a column vector, c: $\begin{matrix}{c = {G^{+}q}} \\{{{where}\mspace{14mu} q} = {{\left\lbrack {\int_{t_{k}}^{t_{k + 1}}{{x(u)}\ {\mathbb{d}u}}} \right\rbrack{and}\mspace{14mu} G} = \left\lbrack {\int_{t_{l}}^{t_{l + 1}}{{g\left( {u - s_{k}} \right)}\ {\mathbb{d}u}}} \right\rbrack}}\end{matrix}$ and where G⁺ denotes the pseudo-inverse of matrix G. 5.The method of recovering an input signal from the output signal of aTime Encoding Machine (TEM) according to claim 1, wherein the TEM ischaracterized at least in part by a relationship δ<(π/Ω)((b−c)/κ), whereδ represents a hysteresis value of the TEM, b represents an outputvoltage of the TEM, κ represents an integration constant of the TEM, Ωrepresents a bandwidth of the input signal x(t) and c represents amaximum amplitude bound of the input signal, and wherein a sequence oftransition times generated by the TEM is expressed as∫_(t_(k))^(t_(k + 1))x(u) 𝕕u = (−1)^(k)[−b(t_(k + 1) − t_(k)) + 2κδ]. 6.A method of recovering a bounded input signal from an output signal of atime encoding machine (TEM) comprising: determining an amplitude andbandwidth of an input signal to the TEM; setting at least one TEMparameter of the Time Encoding Machine in accordance with boundingconstraints based on the determined amplitude and bandwidth of the inputsignal; determining timing of zero crossings indicating state changes ofthe TEM output signal; and applying a non-linear inversion operationbased at least in part on the at least one TEM parameter to recover theinput signal from a subset of zero crossings.
 7. The method of claim 6wherein the TEM is an ASDM including a Schmitt trigger and wherein aparameters related to the TEM output signal is a Schmitt trigger boundlevel.
 8. The method of claim 6 wherein the TEM includes a Schmitttrigger and wherein a parameters related to the TEM output signal is aSchmitt trigger hysteresis level.
 9. The method of claim 6 wherein theTEM is an integrate and fire neuron circuit including threshold detectorand wherein a parameter related to the TEM output signal is a thresholddetector level.
 10. The method of claim 9, wherein the integrate andfire neuron circuit includes an integrator constant and wherein afurther parameters related to the TEM output signal is the integratorconstant.
 11. The method of claim 6, wherein the non-linear inversionoperation comprises a recursive algorithm employing successive zerocrossings.
 12. The method of claim 11, wherein the recursive algorithmis expressed as x(t)=g^(T)(t)[(B⁻¹)(BG(B⁻¹)^(T))⁺[Bq].
 13. The method ofclaim 11, wherein the recursive algorithm is expressed as$\begin{matrix}{{x(t)} = {\begin{matrix}{\lim\;{x_{l}(t)}} \\\left. l\rightarrow\infty \right.\end{matrix} = {{hH}^{+}{q.}}}} \\{{{{where}\mspace{14mu} h} = {\frac{1}{2}\left\lbrack {g\left( {t - t_{k + 1}} \right)} \right\rbrack}^{T}},{q = \left\lbrack {\int_{t_{k}}^{t_{k + 2}}{{x(u)}\ {\mathbb{d}u}}} \right\rbrack}} \\{H = {\frac{1}{2}\left\lbrack {\int_{t_{l}\;}^{t_{l + 2}}{{g\left( {t - t_{K + 1}} \right)}{\mathbb{d}u}}} \right\rbrack}}\end{matrix}$ and wherein H⁺ denotes a pseudo-inverse of H.
 14. A systemfor time encoding and time decoding of a signal x(t) comprising: a timeencoding machine, the time encoding machine having an output voltageswing b, and integrator constant, κ, and a hysteresis value δ, whereinδ<(π/Ω)((b−c)/κ), where Ω represents a bandwidth of the input signalx(t) and c represents a maximum amplitude of the input signal x(t) andwhere |c|<b, the TEM generating a sequence of transition times:∫_(t_(k))^(t_(k + 1))x(u) 𝕕u = (−1)^(k)[−b(t_(k + 1) − t_(k)) + 2κδ]. atime decoding machine, the time decoding machine receiving the sequenceof transition times and applying at least one non-linear operation torecover the input signal x(t) therefrom.
 15. The system of claim 14,wherein the sequence of transition times are exactly known and whereinthe time decoding machine fully recovers the input signal from thesequence of transition times.
 16. The system of claim 14, wherein thesequence of transition times are quantized and wherein precision of therecovery of the input signal by the TDM is related to a number of bitsused to quantize the sequence of transition times.
 17. The system ofclaim 14, wherein the transition times are quantized in the TDM whichincludes an N-bit counter driven by a clock having a clock frequency,the counter outputting an N-bit counter value on each occurrence of atransition time.
 18. The system of claim 14, wherein the TEM is anAsynchronous Sigma Delta Modulator.
 19. A time decoding machine (TDM),the time decoding machine recovering a band limited signal from asequence of transitions generated by a time encoding machine, the TDMcomprising: a time measurement circuit, the time measurement circuitgenerating an N-bit quantized representation of time values of zerocrossings indicating the sequence of transitions provided at an inputterminal of the time measurement circuit; a processor coupled to thetime measurement circuit and receiving output of the time measurementcircuit at a rate at least equal to a Nyquist rate of a signal to bedecoded, the processor being programmed to perform the steps: generatinga set of weighted impulse functions based upon the time values of thezero crossings; and applying the set of weighted impulse functions to animpulse response filter to recover the input signal.
 20. The TDM ofclaim 19, wherein consecutive zero crossings of the TEM output signaldefine time intervals and wherein the weighted impulse functions havetime values that are centered on the time intervals of the TEM outputsignal.
 21. The TDM according to claim 19, wherein the weighted impulsefunctions have a weighting value determined by the processor byevaluating consecutive zero crossings and at least one property of theTEM.
 22. The TDM according to claim 21, wherein the weighted impulsefunctions have weighting value coefficients c_(k) determined by theprocessor which are expressed in matrix form as a column vector, c:c = G⁺qwhere  q = [∫_(t_(κ))^(t_(κ + 1))x(u) 𝕕u]  and  G = [∫_(t_(l))^(t_(l + 1))g(u − s_(k)) 𝕕u]and where G⁺ denotes a pseudo-inverse of matrix G.
 23. The TDM accordingto claim 20, wherein the TEM is characterized at least in part by therelationship δ<(π/Ω)((b−c)/κ), where δ represents a hysteresis value ofthe TEM, b represents an output voltage of the TEM, κ represents anintegration constant of the TEM, Ω represents a bandwidth of the inputsignal x(t) and c represents a maximum amplitude bound of the inputsignal, and wherein the sequence of transition times generated by theTEM is expressed as∫_(tk)^(k + 1)x(u) 𝕕u = (−1)^(k)[−b(t_(k + 1) − t_(k)) + 2k δ].
 24. Thesystem of claim 14, wherein the TEM is an integrate and fire neuroncircuit.